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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Reduction of Order (For reading only)</dfn>Suppose that <span class="process-math">\(y_1(x)\)</span> is a solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0.
\end{equation*}
</div>
<p class="continuation">To find another solution, we let <span class="process-math">\(y_2=v(x) y_1\text{.}\)</span> Substituting it into the ODE,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;v^{\prime \prime} y_1+2 v^{\prime} y_1^{\prime}+v y_1^{\prime \prime}+p(x) (v^{\prime} y_1+v y_1^{\prime})+q(x) v y_1=0\\
&amp; v [y_1^{\prime \prime}+p(x)y_1^{\prime}+q(x) y_1]+v^{\prime \prime} y_1+v^{\prime}[2 y_1^{\prime}+p(x) y_1]=0\\
&amp;\qquad \qquad \qquad \qquad \quad 0+v^{\prime \prime} y_1+v^{\prime}[2 y_1^{\prime}+p(x) y_1]=0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Setting <span class="process-math">\(u=\frac{\textrm{d} v}{\textrm{d} x}\text{,}\)</span> one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_1 \frac{\textrm{d} u}{\textrm{d} x}+[2 y_1^{\prime}+p(x) y_1] u=0.
\end{equation*}
</div>
<p class="continuation">This is a first order ODE for <span class="process-math">\(u\text{,}\)</span> you can solve it by using the integrating factor.</p>
<span class="incontext"><a href="sec3_5.html#p-109" class="internal">in-context</a></span>
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